Question 13 - A-Level Maths

The matrix A represents a reflection in the line \(y = m x\), where \(m\) is a constant. Show that \(\mathbf { A } = \left( \frac { 1 } { m ^ { 2 } + 1 } \right) \left[ \begin{array} { c c } 1 - m ^ { 2 } & 2 m
2 m & m ^ { 2 } - 1 \end{array} \right]\)
You may use the result in the formulae booklet. 13
\(\quad\) The matrix \(\mathbf { B }\) is defined as \(\mathbf { B } = \left[ \begin{array} { l l } 3 & 0
0 & 3 \end{array} \right]\)
Show that \(( \mathbf { B A } ) ^ { 2 } = k \mathbf { I }\)
where \(\mathbf { I }\) is the \(2 \times 2\) identity matrix and \(k\) is an integer.
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1. The diagram below shows a point \(P\) and the line \(y = m x\) Draw four lines on the diagram to demonstrate the result proved in part (b).
  Label as \(P ^ { \prime }\) the image of \(P\) under the transformation represented by (BA) \({ } ^ { 2 }\)
  \includegraphics[max width=\textwidth, alt={}, center]{74b8239a-1f46-45e7-ad20-2dce7bf4baf6-20_579_1068_584_488} 13
(ii) Explain how your completed diagram shows the result proved in part (b).
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The matrix \(\mathbf { C }\) is defined as \(\mathbf { C } = \left[ \begin{array} { c c } \frac { 12 } { 5 } & \frac { 9 } { 5 }
\frac { 9 } { 5 } & - \frac { 12 } { 5 } \end{array} \right]\)
Find the value of \(m\) such that \(\mathbf { C } = \mathbf { B A }\) Fully justify your answer.
[0pt] [4 marks]